A317344 O.g.f. A(x) satisfies: [x^n] exp( n^2*x - n*A(x) ) = 0 for n >= 1.

1, 1, 6, 74, 1400, 35676, 1140328, 43740848, 1954336608, 99561874080, 5691393582336, 360561583177440, 25064797000415744, 1896477768217789120, 155128714525468598400, 13639839907494680219648, 1282811359778733608826368, 128498290985443181787800064, 13657938489514600713859515392, 1535272989503239280608301470720, 181975961346350933380240113192960

Offset

1,3

Comments

It is remarkable that this sequence should consist entirely of integers.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = (2 + LambertW(-2*exp(-2))) * sqrt(-LambertW(-2*exp(-2)) * (1 + LambertW(-2*exp(-2)))) / (4*sqrt(2)*Pi) = 0.0440433939... - Vaclav Kotesovec, Aug 06 2018

Example

O.g.f.: A(x) = x + x^2 + 6*x^3 + 74*x^4 + 1400*x^5 + 35676*x^6 + 1140328*x^7 + 43740848*x^8 + 1954336608*x^9 + 99561874080*x^10 + ...
such that [x^n] exp( n^2*x - n*A(x) ) = 0  for n >= 1.
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^2*x - n*A(x) ) begins:
n=1: [1, 0, -2, -36, -1764, -167280, -25620600, -5737974480, ...];
n=2: [1, 2, 0, -88, -4160, -371328, -55329536, -12201990400, ...];
n=3: [1, 6, 30, 0, -7812, -698184, -97733304, -20791334880, ...];
n=4: [1, 12, 136, 1296, 0, -1171968, -168658176, -33909447168, ...];
n=5: [1, 20, 390, 7220, 113020, 0, -265712600, -55963975600, ...];
n=6: [1, 30, 888, 25704, 709056, 16600320, 0, -84622337280, ...];
n=7: [1, 42, 1750, 72072, 2909340, 112245672, 3684715944, 0, ...];
n=8: [1, 56, 3120, 172640, 9455488, 508540416, 26199517696, 1150524892160, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 3*x^2/2! + 43*x^3/3! + 1945*x^4/4! + 178041*x^5/5! + 26792971*x^6/6! + 5940440563*x^7/7! + ... + A317343(n)*x^n/n! + ...

Prog

(PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); polcoeff( log(Ser(A)), n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A317343, A317346, A317347.

Sequence in context: A232330 A269337 A057783 * A357011 A231691 A177561

Adjacent sequences: A317341 A317342 A317343 * A317345 A317346 A317347

Keyword

nonn

Author

Paul D. Hanna, Jul 26 2018

Status

approved